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Mass Conservation

Mixture of Gases Continuity:

\bgroup\color{blue}$\displaystyle {\frac{{\partial}}{{\partial t}}}$\egroup\bgroup\color{blue}$\displaystyle \left(\vphantom{ \alpha_m\rho_m }\right.$\egroup\bgroup\color{blue}$\displaystyle \alpha_{m}^{}$\egroup\bgroup\color{blue}$\displaystyle \rho_{m}^{}$\egroup\bgroup\color{blue}$\displaystyle \left.\vphantom{ \alpha_m\rho_m }\right)$\egroup + \bgroup\color{red}$\displaystyle {\frac{{\partial}}{{\partial z}}}$\egroup\bgroup\color{red}$\displaystyle \left(\vphantom{ \alpha_m\rho_m V_m }\right.$\egroup\bgroup\color{red}$\displaystyle \alpha_{m}^{}$\egroup\bgroup\color{red}$\displaystyle \rho_{m}^{}$\egroupVm\bgroup\color{red}$\displaystyle \left.\vphantom{ \alpha_m\rho_m V_m }\right)$\egroup = \bgroup\color{magenta}$\displaystyle \sum_{{x=l,s}}^{}$\egroup\bgroup\color{magenta}$\displaystyle \Gamma_{{xg}}^{}$\egroup

Noncondensable Gas Continuity:

\bgroup\color{blue}$\displaystyle {\frac{{\partial}}{{\partial t}}}$\egroup\bgroup\color{blue}$\displaystyle \left(\vphantom{ \alpha_m\rho_m X_n }\right.$\egroup\bgroup\color{blue}$\displaystyle \alpha_{m}^{}$\egroup\bgroup\color{blue}$\displaystyle \rho_{m}^{}$\egroupXn\bgroup\color{blue}$\displaystyle \left.\vphantom{ \alpha_m\rho_m X_n }\right)$\egroup + \bgroup\color{red}$\displaystyle {\frac{{\partial}}{{\partial z}}}$\egroup\bgroup\color{red}$\displaystyle \left(\vphantom{ \alpha_m\rho_m X_n V_m }\right.$\egroup\bgroup\color{red}$\displaystyle \alpha_{m}^{}$\egroup\bgroup\color{red}$\displaystyle \rho_{m}^{}$\egroupXnVm\bgroup\color{red}$\displaystyle \left.\vphantom{ \alpha_m\rho_m X_n V_m }\right)$\egroup = \bgroup\color{green}$\displaystyle {\frac{{\partial}}{{\partial z}}}$\egroup\bgroup\color{green}$\displaystyle \left(\vphantom{ \alpha_m D_n^X \frac{\partial X_n}{\partial z} }\right.$\egroup\bgroup\color{green}$\displaystyle \alpha_{m}^{}$\egroupDnX\bgroup\color{green}$\displaystyle {\frac{{\partial X_n}}{{\partial z}}}$\egroup\bgroup\color{green}$\displaystyle \left.\vphantom{ \alpha_m D_n^X \frac{\partial X_n}{\partial z} }\right)$\egroup + \bgroup\color{green}$\displaystyle {\frac{{\partial}}{{\partial z}}}$\egroup\bgroup\color{green}$\displaystyle \left(\vphantom{ \alpha_m D_n^\rho \frac{\partial \rho_m}{\partial z} }\right.$\egroup\bgroup\color{green}$\displaystyle \alpha_{m}^{}$\egroupDn\bgroup\color{green}$\scriptstyle \rho$\egroup\bgroup\color{green}$\displaystyle {\frac{{\partial \rho_m}}{{\partial z}}}$\egroup\bgroup\color{green}$\displaystyle \left.\vphantom{ \alpha_m D_n^\rho \frac{\partial \rho_m}{\partial z} }\right)$\egroup


Liquid Continuity:

\bgroup\color{blue}$\displaystyle \epsilon_{v}^{}$\egroup\bgroup\color{blue}$\displaystyle {\frac{{\partial}}{{\partial t}}}$\egroup\bgroup\color{blue}$\displaystyle \left(\vphantom{ \alpha_l\rho_l }\right.$\egroup\bgroup\color{blue}$\displaystyle \alpha_{l}^{}$\egroup\bgroup\color{blue}$\displaystyle \rho_{l}^{}$\egroup\bgroup\color{blue}$\displaystyle \left.\vphantom{ \alpha_l\rho_l }\right)$\egroup + \bgroup\color{red}$\displaystyle \epsilon_{v}^{}$\egroup\bgroup\color{red}$\displaystyle {\frac{{\partial}}{{\partial z}}}$\egroup\bgroup\color{red}$\displaystyle \left(\vphantom{ \alpha_l\rho_l V_l }\right.$\egroup\bgroup\color{red}$\displaystyle \alpha_{l}^{}$\egroup\bgroup\color{red}$\displaystyle \rho_{l}^{}$\egroupVl\bgroup\color{red}$\displaystyle \left.\vphantom{ \alpha_l\rho_l V_l }\right)$\egroup = \bgroup\color{magenta}$\displaystyle \sum_{{x=g,s}}^{}$\egroup\bgroup\color{magenta}$\displaystyle \Gamma_{{xl}}^{}$\egroup

Solid Continuity:

\bgroup\color{blue}$\displaystyle \epsilon_{v}^{}$\egroup\bgroup\color{blue}$\displaystyle {\frac{{\partial}}{{\partial t}}}$\egroup\bgroup\color{blue}$\displaystyle \left(\vphantom{ \alpha_s\rho_s }\right.$\egroup\bgroup\color{blue}$\displaystyle \alpha_{s}^{}$\egroup\bgroup\color{blue}$\displaystyle \rho_{s}^{}$\egroup\bgroup\color{blue}$\displaystyle \left.\vphantom{ \alpha_s\rho_s }\right)$\egroup = \bgroup\color{magenta}$\displaystyle \sum_{{x=g,l}}^{}$\egroup\bgroup\color{magenta}$\displaystyle \Gamma_{{xs}}^{}$\egroup



Michael L. Hall